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Quasiclassical expressions for the free energy of superconducting systems

In the seminal work by G. Eilenberger [Z. Phys. 214, 195 (1968)], the quasiclassical expression for the free energy of spin-singlet superconductor has been suggested. Starting from the Luttinger-Ward formulation we derive the Eilenberger free energy and find its generalization for superconductor or superfluid with spin-triplet correlations. Besides ordinary superconductors with various scattering mechanisms, the obtained free energy functional can be used for systems with spin-triplet pairing such as superfluid $^3$He and superconducting systems with spatially-inhomogeneous exchange field or spin-orbit coupling. Using this general result we derive the simplified expression for the free energy in the diffusive limit in terms of the momentum-averaged propagators.